4 ms·

I had a very similar moment when I was thinking about why the determinants of all (non-identity) projections are 0 and the fact that the determinant is the prod

• deleted 6y ago
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• Balgair 6y ago
  > the determinant is the product of the eigenvalues. Ohhhhh, now I get determinants. I just thought they were some easy way to classify a matrix, not that they had any special meaning. Not that they were also all the eigenvalues multiplied too.
  • jcranmer 6y ago
    And the trace is the sum of the eigenvalues.
  • llamaz 6y ago
    That's one concrete way to view determinants, but it's not _the_ way to think about them. Determinants are kind of abstract and even inelegant. This is because it's defined in terms of coordinates rather than abstractly. A coordinate free definition exists, but it's very complicated. Ideally they would not be taught at all, but it's useful for doing exam problems. Cramer's rule in particular is quite useful when solving simultaneous equations in engineering under exam conditions, for example. Even in computer/engineering applications, row reduction/diagonal normal form type are used instead. Determinants are a way to get a scalar number out of a matrix. You have linear forms, bilinear forms, and multilinear forms, the latter of which determinants fall under. There are certain properties you would like the determinant to satisfy, such as det(AB) = det(A) det(B), so that the determinant can give you an idea of whether a matrix is invertible. Eventually you work out that a determinant _has to be_ a multilinear form with certain properties (alternating, skew symmetric). Again this is all very complex and messy. Initially 'linear algebra' was just the study of determinants, because people thought of math in terms of solving equations. Linear algebra, as it's known today, came later. Bilinear forms turn up all the time, so ideally you would first learn bilinear forms, then learn the properties a bilinear form can have (e.g. alternating, skew symmetry, etc), and then learn determinants in that context.
    • mkl 6y ago
      Determinants aren't inherently abstract, they're quite practical, especially in the article's context: they tell you the scaling factor of the transformation the matrix represents, and whether it does a reflection. E.g. the determinants of the rotation matrices in the article are all 1, because they don't scale or reflect anything, just rotate.
      • MiroF 6y ago
        The matrix {{1,0,0},{0,-1,0},{0,0,-1}} has a determinant of 1 too, but it reflects along two different axes. So there's no way to distinguish from the determinant between the identity and between a reflection across two axes. I think the important detail is if you're changing from right-handed bases to left-handed or vice versa during the transformation.
        • rumanator 6y ago
          > So there's no way to distinguish from the determinant between the identity and between a reflection across two axes. I fail to see the point you tried to make. Both transformations preserve volume. Why did you expected volume to change if you rotate something?
          • MiroF 6y ago
            Parent comment claims you can use determinant to see if a reflection is occurring, I disagreed. I don't know where you thought I expected volume to change if I rotate something.
        • mkl 6y ago
          Two reflections make a rotation. Your matrix is a rotation matrix that does no reflection (everything will be oriented the same way after being transformed).
          • MiroF 6y ago
            You're right, cheers.